It is worth noting that in special cases such as two-type models described above, it is possible to derive exact solutions [1]}. From a numerical perspective, terms such as hypergeometric functions arising in these such solutions can present numerical challenges in some parameter regimes [2]}. When this is the case, which often occurs when these solutions are required within iterative schemes, it can be preferable to solve \(\phi _1\)  using numerical differential equation solvers. Moreover, the numerical recipe we advocate above for the solving the system holds for any \(m\) -type model in our class.
